Magnetically induced ferroelectricity in orthorhombic manganites: Microscopic origin and chemical trends

The microscopic origin of the magnetically driven ferroelectricity in collinear $E$-type antiferromagnetic (AFM-$E$) orthorhombic manganites is explained by means of first-principles Wannier functions. We show that the polarization is mainly determined by the asymmetric electron hopping of orbitally polarized $e_g$ states, implicit in the peculiar in-plane zigzag spin arrangement in the AFM-$E$ configuration. In $ortho{\text{-HoMnO}}_3$, Wannier-function centers are largely displaced with respect to corresponding ionic positions, implying that the final polarization is strongly affected by a purely electronic contribution, at variance with standard ferroelectrics where the ionic displacement is dominant. However, the final value of the polarization is the result of competing effects, as shown by the opposite signs of the contributions to the polarization coming from the $\text{Mn}{e}_g$ and $t_{2g}$ states. Furthermore, a systematic analysis of the link between ferroelectricity and the spin, orbital, and lattice degrees of freedom in the manganite series has been carried out, in the aim of ascertaining chemical trends as a function of the rare-earth ion. Our results show that the Mn-O-Mn angle is the key quantity in determining the exchange coupling: upon decreasing the Mn-O-Mn angle, the first- (second-) nearest-neighbor ferromagnetic (antiferromagnetic) interaction decreases (remains constant), in turn stabilizing either the $A$-type antiferromagnetic or the AFM-$E$ spin configuration for weakly or strongly distorted manganites, respectively. The $\text{Mn}{e}_g$ contribution to the polarization dramatically increases with the Mn-O-Mn angle and decreases with the “long” Mn-O bond length, whereas the $\text{Mn}{t}_{2g}$ contribution decreases with the “short” Mn-O bond length, partially canceling the former term.

Figure 1

(Color online) Atomic displacements in FE ${\text{HoMnO}}_3$, as obtained by the difference of atomic coordinates in optimized AFM-$E$ and optimized AFM-$A$ spin configurations (length of arrows in arbitrary units). In the ${\text{MnO}}_2$ plane, the AFM-$E$ spin arrangement is also shown with black (up) and white (down) Mn atoms. A schematic representation of the $e_g$ orbital ordering and the direction of the ferroelectric polarization are also shown. The axis $X$ corresponds to $a$ $\left(b\right)$ and $Z$ corresponds to $c$ $\left(a\right)$ axis in the $Pnma$ $\left(Pbnm\right)$ setting.

Figure 2

Energy band structure of optimized AFM-$E$ ${\text{HoMnO}}_3$. Due to the AFM-$E$ state, only up-spin channel is plotted. $E_F$ denotes the Fermi energy and it is set as zero of the energy scale.

Figure 3

(Color online) Isosurface of WFs for $\text{Mn}d$ states centered at Mn(0) site in AFM-$E$ ${\text{HoMnO}}_3$: (a) $e_g$ states and (b) $t_{2g}$ states. The green arrows indicate the displacement of the centers of WFs from the Mn atomic position. Here, the superscript p (ap) of O denotes oxygen ion between parallel (antiparallel) spin of Mn ions.

Figure 4

Schematic representation of the mechanism which causes microscopic polarization of $\text{Mn}{e}_g$ orbital due to the asymmetric electronic hopping. Bottom sketch: Hopping is allowed only from Mn(0) to Mn(1) and not to Mn(3): this determines the direction of WFC (empty arrow). To increase hopping, Mn(0) moves “right” (see filled arrow), giving a lower weight of WF on Mn(0) but larger on Mn(1) [represented as dashed vs solid (before vs after displacement) Gaussians on the atoms]. Also shown is the resulting P (see arrow). Top sketch: Resulting atomic displacement (shown as filled arrows) aimed at increasing ${\varphi }^p$.

Figure 5

(Color online) Isosurface of WFs of $\text{O}2p$ state. Only four typical WFs in the $ac$ plane are shown. (a) ${\text{O}}^{\text{p}}{p}_y$, (b) ${\text{O}}^{\text{p}}{p}_z$, (c) ${\text{O}}^{\text{ap}}{p}_y$, and (d) ${\text{O}}^{\text{ap}}{p}_z$ (up-spin state). The arrows indicate the displacement of the centers of WFs from the O atomic position.

Figure 6

(Color online) (a) Mn-O bond distance of $R{\text{MnO}}_3$ as a function of $r_R$ (radius of rare-earth ion $R^{3+}$). Our results are shown as round symbols: open and filled circles show long and small Mn-O distances in $ac$ plane, respectively, and the half-filled symbols show a middle Mn-O distance along the $b$ axis. The lines are plotted for an eye guide. (b) Mn-O-Mn angles of $R{\text{MnO}}_3$ as a function of the radius of rare-earth atom. The open symbol shows interplane Mn-O-Mn angles (with apical O along the $b$ axis), whereas the closed symbol shows in the $ac$ plane Mn-O-Mn angles. In both panels, results obtained using $\text{GGA}+U$ (with dashed lines and down-triangles) and $\text{LDA}+U$ (with right-triangles only at ${\text{LaMnO}}_3$) are also shown. The experimental data are also shown for comparison: square from Ref. 14, diamond from Ref. 12, left triangle from Ref. 11, and upper triangle from Ref. 13.

Figure 7

(Color online) (a) Total energy of AFM phases with respect to FM phase. (b) Superexchange energy $\left(J_{ij}\right)$: The first nearest-neighbor in plane $J_{\parallel }^{\text{nn}}$, next-nearest-neighbor in plane $J_{\parallel }^{\text{nnn}}$, first nearest-neighbor inter plane $J_{\perp }^1$, and next-nearest-neighbor interplane $J_{\perp }^2$. (c) Sum of $J_{ij}$ for the AFM-$A$ and AFM-$E$ phases.

Figure 8

(Color online) The Mn-O bond length and Mn-O-Mn bond angle as a function of $r_R$ in $R{\text{MnO}}_3$ structure optimized in AFM-$E$ phase. Here, the superscript p (ap) denotes parallel (antiparallel) spin of Mn ions.

Figure 9

(Color online) The orbital-decomposed partial DOS of $\text{Mn}3d$ components for $R{\text{MnO}}_3$, with $R$ as (a) La, (b) Nd, (c) Ho, and (d) Lu. The Arrows indicate energy gap $\left(E_g\right)$, crystal-field splitting $\left(E_{\text{CF}}\right)$, $e_g$ bandwidth $\left(w\right)$, and on-site exchange interaction energy $\left(J\right)$. The crystal structure was optimized imposing the AFM-$E$ spin configuration. The local frame in the ${\text{MnO}}_6$ octahedron is chosen as $x$: middle (interplane), $y$: short axis, and $z$: long axis.

Figure 10

(Color online) Ferroelectric polarization calculated by BP method, WF, and PCM as a function of $\varphi$ in AFM-$E$ $R{\text{MnO}}_3$ (see text for details). The in-plane Mn-O-Mn bond angle $\varphi$ is averaged between ${\varphi }_p$ and ${\varphi }_{\text{ap}}$.